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6.The Theory of Simple Gases 1.An Ideal Gas in a Quantum Mechanical Microcanonical Ensemble 2.An Ideal Gas in Other Quantum Mechanical Ensembles 3.Statistics of the Occupation Numbers 4.Kinetic Considerations 5.Gaseous Systems Composed of Molecules with Internal Motion 6.Chemical Equilibrium
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6.1.An Ideal Gas in a Quantum Mechanical Microcanonical Ensemble N non-interacting, indistinguishable particles in V with E. ( N, V, E ) = # of distinct microstates Let be the average energy of a group of g >> 1 unresolved levels. Let n be the # of particles in level . Let W { n } = # of distinct microstates associated with a given set of { n }. Let w(n ) = # of distinct microstates associated with level when it contains n particles.
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Bosons ( Bose-Einstein statistics) : See § 3.8 Fermions ( Fermi-Dirac statistics ) : w(n ) = distinct ways to divide g levels into 2 groups; n of them with 1 particle, and g n with none.
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Classical particles ( Maxwell-Boltzmann statistics ) : w(n ) = distinct ways to put n distinguishable particles into g levels. Gibbs corrected
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Method of most probable value ( also see Prob 3.4 ) n * extremize Lagrange multipliers
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BE FD
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BE FD Most probable occupation per level MB
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BE FD MB:
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6.2.An Ideal Gas in Other Quantum Mechanical Ensembles Canonical ensemble : Ideal gas, = 1-p’cle energy : g{ n } = statistical weight factor for { n }.
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Maxwell-Boltzmann : multinomial theorem
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partition function (MB) grand partition function (MB)
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Bose-Einstein / Fermi-Dirac : Difficult to evaluate (constraint on N )
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B.E. F.D. Grand potential : BE FD q potential :
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BE FD MB : c.f. §4.4 Alternatively
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Mean Occupation Number For free particles : BE FD see §6.1
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6.3. Statistics of the Occupation Numbers BE FD Mean occupation number : MB : FD : BE :B.E. condensation Classical : high T must be negative & large From §4.4 : same as §5.5
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Statistical Fluctuations of n BE FD
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BE FD above normal below normal Einstein on black-body radiation : +1 ~ wave character n 1 ~ particle character see Kittel, “Thermal Phys.” Statistical correlations in photon beams :see refs on pp.151-2
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Probability Distributions of n Let p (n) = probability of having n particles in state of energy . BE FD
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BE : BE FD FD :
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MB :Gibbs’ correction Poisson distribution Alternatively “normal” behavior of un-correlated events
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BE : FD : Geometric ( indep of n ) > MB for large n : Positive correlation < MB for large n : Negative correlation
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n - Representation Let n = number of particles in 1-particle state . Non-interacting particles : State of system in the n- representation :
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6.4.Kinetic Considerations From § 6.1 BE FD Free particles :
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BE FD Let p( ) be the probability of a particle in state . Then s = 1 : phonons s = 2 : free p’cles All statistics
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pressure is due to particle motion (kinetics) Let n f(u) d 3 u = density of particles with velocity between u & u+du. # of particles to strike wall area dA in time dt = # of particles with u dA >0 within volume u dA dt Total impulse imparted on dA = Each particle imparts on dA a normal impluse =
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Rate of Effusion Rate of gas effusion per unit area through a hole in the wall is # of particles to strike wall area dA in time dt All statistics R u Effused particles more energetic. u > 0 Effused particles carry net momentum (vessel recoils) Prob. 6.14
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6.5.Gaseous Systems Composed of Molecules with Internal Motion Assumptions ( ideal Boltzmannian gas ) : 1. Molecules are free particles ( non-interacting). 2. Non-degeneracy (MB stat) : = quantum # for internal DoF
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Internal DoF Molecules : Homopolar molecules (A-A) :
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6.5.A. Monatomic Molecules Let ( All atoms are neutral & in electronic ground state ) Nuclear spin Hyperfine structure : T ~ 10 1 – 10 0 K. Level-splitting treated as degeneracy : Inert gases ( He, Ne, Ar,... ) : Ground state L = S = 0 : = 0 denotes ground state. 0 = 0. L = 0; S 0 :
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L = 0, S 0
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L 0, S 0 Ground state 0 = 0. C V, int = 0 in both limits C V has a maximum.
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6.5.B. Diatomic Molecules Let ( All atoms are neutral & in electronic ground state ) Non-degenerate ground state ( most cases ) g e = 1& j elec (T) = 1 Degenerate ground state ( seldom ) : 1. Orbital angular momentum 0, but spin S = 0 : In the absence of B, depends on | z | doublet ( z = M ) is degenerate ( g e = 2 = j(T) ) C V = 0 2. = 0, S 0 : g e = 2S + 1 = j(T) C V = 0 3. 0 & S 0 : Spin-orbit coupling B eff fine structure
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E.g., NO ( 1/2, 3/2 ) ( splitting of doublet ) : C V has max. for some kT ~
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Vibrational States for diatomic gases Full contribution for T 10 4 K No contribution for T 10 2 K Harmonic oscillations (small amplitude) : From § 3.8 : equipartition value vib DoF frozen out
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Very high T anharmonic effects C vib T( Prob 3.29-30)
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Nuclear Spin & Rotational States: Heteropolar Molecules Heteropolar molecules ( AB ) : no exchange effects interaction between nuclear spin & rotational states negligible. From § 6.5.A : C nucl = 0 Molecule ~ rigid rotator with moment of inertia ( bond // z-axis ) = reduced mass r 0 = equilibrium bond length
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Homopolar molecules
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6.6.Chemical Equilibrium
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